Fundamental Quantum Optics Experiments Conceivable With Satellites -- Reaching Relativistic Distances and Velocities 1206.4949

7/27/2019 Fundamental Quantum Optics Experiments Conceivable With Satellites -- Reaching Relativistic Distances and Velocities 1206.4949

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Fundamental quantum optics experiments conceivable with satellites reaching relativistic distances and velocities

David Rideout 1 , 2 ,3 , Thomas Jennewein 2 , 4 , Giovanni Amelino-Camelia 5 , Tommaso FDemarie 6 , Brendon L Higgins 2 ,4 , Achim Kempf 2 ,3 , 4 , Adrian Kent 3 ,7 ,Raymond Laamme 2 ,3 , 4 , Xian Ma 2 ,4 , Robert B Mann 2 ,4 , Eduardo Martn-Martnez 2 ,4 ,Nicolas C Menicucci 3 ,8 , John Moffat 3 , Christoph Simon 9 , Rafael Sorkin 3 , Lee Smolin 3 ,Daniel R Terno 61 current address: Department of Mathematics, University of California / San Diego, La Jolla, CA, USA

E-mail: [email protected] Institute for Quantum Computing, Waterloo, ON, Canada E-mail: [email protected] Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada4 University of Waterloo, Waterloo, ON, Canada5 Departimento di Fisica, Universit` a di Roma La Sapienza, Rome, Italy6 Department of Physics and Astronomy, Macquarie University, Sydney, NSW, Australia7 Centre for Quantum Information and Foundations, DAMTP, University of Cambridge, Cambridge, U.K.8 School of Physics, The University of Sydney, NSW, Australia9 University of Calgary, Calgary, AB, Canada

Abstract. Physical theories are developed to describe phenomena in particular regimes, and generally are valid onlywithin a limited range of scales. For example, general relativity provides an effective description of the Universe at largelength scales, and has been tested from the cosmic scale down to distances as small as 10 meters [1, 2]. In contrast,quantum theory provides an effective description of physics at small length scales. Direct tests of quantum theory havebeen performed at the smallest probeable scales at the Large Hadron Collider, 10 20 meters, up to that of hundredsof kilometers [3]. Yet, such tests fall short of the scales required to investigate potentially signicant physics that arisesat the intersection of quantum and relativistic regimes. We propose to push direct tests of quantum theory to larger andlarger length scales, approaching that of the radius of curvature of spacetime, where we begin to probe the interactionbetween gravity and quantum phenomena. In particular, we review a wide variety of potential tests of fundamental physicsthat are conceivable with articial satellites in Earth orbit and elsewhere in the solar system, and attempt to sketch themagnitudes of potentially observable effects. The tests have the potential to determine the applicability of quantum

theory at larger length scales, eliminate various alternative physical theories, and place bounds on phenomenologicalmodels motivated by ideas about spacetime microstructure from quantum gravity. From a more pragmatic perspective,as quantum communication technologies such as quantum key distribution advance into Space towards large distances,some of the fundamental physical effects discussed here may need to be taken into account to make such schemes viable.

a r X i v : 1 2 0 6 . 4 9 4 9 v 1 [ q u a n t - p h ] 2 1 J u n 2 0 1 2


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CONTENTS 2

Contents

1 Introduction 31.1 Classication of experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Entanglement tests 7

2.1 Tests of local realism The Bell test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Long distance Bell test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Bell test with human observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Bell test with detectors in relative motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Bell experiments with macroscopic amplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 Bimetric gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

3 Relativistic effects in quantum information theory 123.1 Special relativistic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

3.1.1 Lorentz transformations and polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.2 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 General relativistic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.1 Relativistic frame dragging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.2 Entanglement in the presence of curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 The Fermi problem and spacelike entanglement tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 COW experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

4 Tests of quantum eld theory in non-inertial frames 194.1 Test of the Unruh effect, entanglement delity and acceleration . . . . . . . . . . . . . . . . . . . . . . . 194.2 Gravitationally induced entanglement decorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Probe the spacetime structure by spacelike entanglement extraction . . . . . . . . . . . . . . . . . . . . 22

5 Quantum gravity experiments 235.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .235.2 Lorentz invariant diffusion of polarization from spacetime discreteness . . . . . . . . . . . . . . . . . . . 24

5.2.1 Lorentz invariant diffusion of CMB polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.2.2 Testing for spacetime discreteness with satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3 Decoherence and spacetime noncommutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.4 Relativity of Locality from doubly special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 Quantum communication and cryptographic schemes 256.1 Quantum cryptography with satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2 Quantum tagging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .266.3 Quantum teleportation with satelli tes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7 Techniques which can be used to gain accuracy or isolate certain effects 277.1 Lorentz invariant encodings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.2 Preparation contextuality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

8 Technology 288.1 Measuring the new effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .288.2 Eliminating known sources of noise from the signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.3 Proposed systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

9 Conclusion 30

10 Author contributions 30

11 Acknowledgements 30

12 References 30


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CONTENTS 3

1. Introduction

Our knowledge is ultimately restricted by the boundaries of what we have explored by direct observation or experiment.Experiments conducted within previously inaccessible regimes have often revealed new aspects of the Universe,facilitating new insights into its fundamental operation. Examples of this pervade the history of physical science.Recently, the theories of general relativity and quantum eld theory have arisen to describe aspects of the Universethat can only be accessed experimentally within the regimes of the very large and the very small, respectively. Thespectacular array of useful technologies brought about owing to the formulation of these theories over the previouscentury are vociferous testament to the utility of expanding our experimental horizons.

The success of these two theories also confronts us with a formidable challenge. On one hand, quantum theoryexcellently describes the behaviour of physical systems at small length scales. On the other hand, general relativitytheory excellently describes systems involving very large scales: long distances, high accelerations, and massive bodies .Each of these theories has successfully weathered copious experimental tests independently, yet the two theories arefamously incompatible in their fundamental assertions. One expects that both theories are limiting cases of one set of overarching laws of physics. However, the tremendous experimental success of quantum theory and general relativity,i.e., their enormous individual ranges of validity, makes it extremely difficult to nd experimental evidence that pointsus towards such unifying laws of physics a fully quantum theory of gravity.

Theoretical research has yielded intriguing indications about this sought-after unifying theory of quantum gravity.For example, studies of quantum effects in the presence of black holes, such as Hawking radiation and evaporation,indicate that it will be crucial to understand how the ow and transformations of information are impacted by relativisticand quantum effects, with the notion of entanglement playing a central role.

Ultimately, however, the eld of quantum gravity will require more experimental guidance. So far, the best potentialfor solid experimental evidence for full-blown quantum gravitational effects stems from observations of the cosmicmicrowave background (CMB). However, while of the highest interest, the observation of quantum gravity effects in theCMB would still constitute only a passive one-shot experimental observational opportunity we cannot repeat the bigbang.

In this paper we thus envisage possible avenues for active experimental probes of quantum phenomena at largelength scales, towards those at which gravitational effects will play an increasingly signicant role. The aim in the shortterm is to probe more-or-less solid theoretical expectations, while in the longer term to explore physical regimes in whichthe predictions of theory are not as clear.

To begin making inroads, it seems necessary to test the behaviour of quantum systems, particularly those withentanglement, while these systems possess high speeds and are separated by large distances. On Earth, tests of quantumentanglement have been performed at distances up to 144 km [ 3]. While this is a signicant achievement, it falls short of the large scale relevant for relativistic considerations. Additionally, these tests were performed with stationary detectors.While it is difficult to perform tests with moving detectors, it is conceivable to measure entanglement with beamsplittersmoving at up to 1000 m/s ( 10 6 c) [5]. However, this also falls short one would need to perform entanglement testsin which the detectors are in relative motion at speeds nearer lightspeed ( c) where relativistic effects become signicant.It is interesting to note that active laboratory measurements of gravity at small scales using atomic interferometers havealso been proposed [1].

Quantum repeater networks are a promising candidate for the long distance dissemination of quantumentanglement [6]. However, the study of quantum repeaters shows that even with optimistic estimates, reaching 1000 kmwill be a huge challenge. Even if quantum transmission technology is developed that is capable of transmitting entangledsystems around the Earth, the maximum separation between detectors that can be achieved is bounded by the Earthsdiameter: around 13,000km.

To achieve tests at greater distances and speeds, one needs to move off of planet Earth, and into Space. It

is conceivable that in the not-too-distant future one could perform quantum entanglement tests at the scale of inter-planetary distances, with the associated velocities. For the nearer term, the next step is to perform quantum experimentsthat utilize Earth-orbiting satellite platforms. A satellite in low Earth orbit (LEO), for example, will allow distancesgreater than 10 6 m and relative speeds of two detectors of 10 5 c.

Satellite missions for quantum communications have been considered in various congurations [7, 8, 9, 10, 11], andsome scientic tests utilizing such satellites have been proposed. The Canadian Space Agency (CSA) and the Institutefor Quantum Computing (IQC) have been participating in ongoing studies, dubbed QEYSSAT [12], emphasizing the

At the very smallest of physical scales the Planck scale one expects the gravitational interaction to become comparable to all others,so that quantum and gravitational effects are both simultaneously manifest. The same may also occur at the largest physical scale, that of the cosmos as a whole, wherein quantum effects may account for formation of large scale structures and the cosmic acceleration [4].


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Explorable Scales

2x10 5 m Ground based Experiments

1x10 6 m LEO Satellite: Bell Test, QKD, MovingObservers, Quantum Networks

3.6x10 7 m GEO Satellite: Bell Test, QKD

3.8x10 8 mlight-second Moon: Bell Test, Human Observers

Envisioned;feasibility limit forquantum optics

tests, using futuretechnology

5 - 40x10 9 mSolar Orbit, Mars: Bell Tests, Human

Observers, Long-Range Entanglement

Solar system: using spacecraft totest Quantum Gravity or Long-Range

Entanglement.

> 10 12 m

Beyond our Solar System: Tests ofQuantum Gravity?

> 10 13 mlight-hours

Current;ground based

0.3 km/s

8 km/s

3 km/s

1 km/s

Near Term;possible with

today'stechnology

Long Term;current limit of

todays quantumoptics technology

Figure 1. Overview of the distance and velocity scales achievable in a space environment explorable with man-madesystems, with some possible quantum optics experiments at each given distance.

wider and long term context of such missions for science as well as quantum applications such as global-scale quantumkey distribution (QKD).

These and other proposals for satellite-based quantum apparatuses open a door to distances and velocities thatare either prohibitively impractical or simply impossible to achieve on the ground. Here we describe a number of ideasstemming from a series of discussions that took place at the Perimeter Institute for Theoretical Physics, which focusedon what tests of fundamental quantum physics could be achieved with such experimental setups. We consider a varietyof scenarios, illustrated in Figure 1, of which some will be accessible with todays technologies, such as a single satelliteat LEO altitudes (5001000km). Experiments at larger distances will be possible only on a longer time frame, owing totheir complexity and the advanced technologies that are required, and include systems in geostationary (GEO) orbits(36,000 km) or even Earth-Moon distances (380,000km). Visionary experiments involving distances at the scale of theEarths distance to the Sun (150Gm) are conceivable, but require technologies yet to be developed.

Here we consider experimental scenarios that are visionary in nature, focusing on the scientic novelty of such


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CONTENTS 5

experiments and their capacity to offer signicant new insights, and intentionally avoiding dwelling on technological ornancial limitations. It follows that these scenarios may be achievable within varying timeframes. The space sciencecommunity may at some point consider these proposals within the context of future missions. Furthermore, we attemptto avoid bias with regard to expectations of experimental outcomes by also considering several ideas which are based onunconventional physical theories.

For each proposed experiment we explain the basic idea and physical concepts that it probes, including referencesfor further details. We also attempt to provide some characterization of the magnitude of the expected effect. Wewish to note that this paper resembles a review article, as it covers a wide spectrum of physical effects, contributed bymultiple authors from a number of perspectives. In order to give a better indication of the primary contributor for eachsection, we indicate the primary authors at the end of the paper.

We hope to encourage physicists to take one or more of these experiment concepts and work towards establishingthe details of how it may work in practice, and more carefully examine the magnitude of observable effects that may beobserved. Similar questions have already been posed for the Space-QUEST project, which aims to place an entangledphoton source on the International Space Station [ 13, 14]. Our analysis goes beyond that work and studies variouspossible science experiments on a broader scope.

The paper is organized as follows. In Section 1.1 we dene a broad classication scheme for the proposedexperiments. In Section 1.2 we present a table summarizing the list of experiments, including some indication of the practical feasibility of each. The experiments themselves are organized into broad categories based upon thenature of the physical theories which the experiment is designed to probe. In Section 2 we discuss EPRB-type testsof the Bell inequalities. In Section 3 we consider the effect of both special and general relativity. In Section 4 wediscuss tests of quantum eld theory in accelerated frames. Section 5 considers possible effects motivated by quantumgravity. In Section 6 we include some experiments whose motivations are directed towards developing useful quantumcommunication technologies, such as quantum key distribution and quantum teleportation. Section 7 considers somevariants on the usual EPRB-type setup which may be useful to consider for a number of the Bell-test scenarios. Finally,in Section 8 we provide some technical details on the current state of the art for quantum optics experiments in Space,and make some concluding remarks in Section 9.

1.1. Classication of experiments

In order to give the reader some initial indicator of the nature of a proposed experiment, we introduce a broadclassication scheme that roughly characterizes the feasibility and ambitiousness of each test. The classes are denedas follows:

Level-1 experiments verify well-established physics in a new regime, often at larger length scales than have yet beenprobed experimentally. Level-2 experiments test physics that is somewhat less certain. An example is an experiment whose predictedoutcome involves the parallel transport of spins in curved spacetime. It seems pretty clear along which spacetime

path the spins should be transported, however physical phenomena whose outcomes involve such computations havenot yet been tested experimentally.

Level-3 experiments consider situations in which the scale of a test is expanded into regimes in which variousproposed alternative theories predict an outcome other than that predicted by conventional physics. The intent of such an experiment is to seek evidence for or against such an alternative theory.

Level-4 experiments test physics in regimes for which there is not yet a standard theory which can be used to predictthe outcome. Instead we propose tests based on models which nd their motivation from the expected (or guessed)nature of physics in these regimes. Tests of quantum gravity fall into this latter category.

1.2. Summary

We consider the experiments summarized in Table 1. For each experiment we indicate a characteristic length scale of the test, the timeframe in which such an experiment may become technically feasible, the domain of physics which isexplored, a level classication as described in Section 1.1, and some indication of the magnitude of a predicted effect.In each case we put the smallest scale experiment considered many experiments can be performed at larger scales aswell.


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CONTENTS 6Name Scale Timeframe Regime Level Observability SectionEntanglement TestsLong distance Bell-test LEO and

beyondnear-term Standard

QM1 Observable 2.2

Bell-test with human ob-servers

Earth-Moon

long-term QM and free-will

3 Need human onMoon surface orlunar orbit

2.3

Detectors in relative mo-tion

LEO mid-term Standard SR 3 achievable 2.4

Amplied entanglement LEO nearmid-term QM 3 potentially achiev-able

2.5

Bimetric gravity LEO near-term Test non-standardtheory

3 bound parameter? 2.6

Special and General Relativistic EffectsLorentz transformed polar-ization

LEO andbeyond

mid-term QM + SR 1 Beyond current tech. 3.1.1

Relativistic frame dragging TBD TBD QM + GR 2 Beyond current tech. 3.2.1Entanglement with curva-

ture

TBD visionary QM + GR 2 Beyond current tech. 3.2.2

Fermi problem Sunshieldedsatellites

long-term QFT 2 Borderline 3.3

Optical Colella-Overhau-ser-Werner experiment

LEO andbeyond

near-term QM + GR 1+ Observable 3.4

Accelerating Detectors in Quantum Field TheoryAcceleration induced -delity loss

TBD visionary QFT + GR 2 Beyond current tech. 4.1

Berry phase interferometry LEO mid-term QFT + GR 2 Borderline 4.1Gravitationally inducedentanglement decorrela-tion

LEO andbeyond

near-term Non-standardQFT + GR

3 Possibly observable 4.2

Spacelike entanglement ex-

traction

TBD visionary QFT + GR 2+ Beyond current tech. 4.3

Quantum Gravity ExperimentsDiffusion of polarization TBD, solar

system?visionary? QG 4 Bound model param-

eters5.2

Spacetime noncommuta-tivity

TBD unknown QG 4 unknown 5.3

Relativity of locality TBD, solarsystem?

unknown QG 4 unknown 5.4

Quantum Communication and Cryptographic SchemesQuantum tagging LEOGEO near-term QM + SR 1 feasible with current

technology?6.2

Quantum teleportation LEO near-term StandardQM

1 feasible with currenttechnology?

6.3

Table 1. Summary of possible experiments. LEO refers to Low Earth Orbit, an elliptical orbit about the Earth withaltitude up to 2000 km. The timeframes are mentioned in Section 1. Roughly, near-term experiments ( 5 years) can beaccomplished with a single satellite in LEO, mid-term experiments (25 years) require multiple satellites or higher orbits,long-term experiments involve Earth-Moon distances, and visionary experiments extend to solar orbits and beyond.Under Regime (and throughout the paper) QM refers to ordinary quantum mechanics, QFT to quantum eld theory,SR to special relativity, GR to general relativity, and QG to quantum gravity. The Level classications are explainedin Section 1.1.


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2. Entanglement tests

The space environment opens the possibility of performing entanglement experiments over extremely long distances,allowing us to push the verication bounds of quantum theory. The possibility of, thereby, observing deviations frompredictions of the theory is tantalizing.

2.1. Tests of local realism The Bell test

Quantum theory tells us that, within a multipartite entangled system, the measurement-induced collapse of the statecaused by measuring one particle will be instantaneously reected in measurement outcomes on the other particles,regardless of how far apart those particles may be. The troubling nature of this, rst realized by Einstein, Podolsky,and Rosen [15] (EPR), was later cast into the form of bipartite spin measurements by Bohm [16] (EPRB), characterizedrigorously by Bell [17], and later again cast into an experimentally-testable manner by Clauser, Horne, Shimony, andHolt (CHSH) [18]. Bell and CHSH each produced an inequality relating the statistical correlations of the outcomesof measurements performed on the two particles under two naturally intuitive assumptions: (1) that physical systemspossess only objective locally-dened properties (independent of measurement context), and (2) that inuences betweensystems cannot propagate faster than light-speed. Under certain congurations, quantum mechanics violates these Bellinequalities.

Experimental tests of Bell inequalities involve gathering correlation statistics by measuring numerous entangledphoton pairs. Thus far, these Bell tests have been consistent with quantum theory, and therefore, despite their

intuitiveness, the assumptions of Bells derivation do not hold.

2.2. Long distance Bell test

The rst experiment we consider involves testing the phenomenon of quantum entanglement over large distances.Quantum mechanics does not predict any breakdown in its description of largely extended quantum states. How validis this assumption? Current experiments on the ground have reached 144 km, and distances beyond that are difficult toimpossible to be explored on the ground. (There is also the possibility that quantum entanglement might be relevant atcosmological scales [19], however it is not clear if observations or experiments are possible to measure such entanglement.See Section 4.3 for some discussion on this issue.)

Such an experiment would also have interest from the quantum foundations point of view: performing a long-distance Bell test with spacelike separated observers will lead to a paradoxical situation for some interpretations of QMwhen one considers the problem of measurement. For two spacelike separated events the concept of simultaneity is

frame dependent and external observers in relative motion with respect to the experiment would answer differently tothe question who measured rst?As discussed in the work of Aharonov and Albert in the context of quantum mechanics [ 20], and later by Sorkin

in the context of quantum eld theory [21], assuming that quantum state reduction upon measurement takes placeon hyperplanes can lead to causality paradoxes. However, to the authors knowledge, an experiment testing for sucha violation of causality has not yet been performed. Also, these results leave open other plausible hypotheses aboutquantum state reduction in particular, that its effects propagate causally [22, 23]. We describe below (in Section 2.5)experiments testing this possibility. The experimental challenges are achieving relativistic relative velocities with respectto the EPR experiment. The more spacelike separated Alice and Bob are, the easier the simultaneity test is to perform.Satellite-based experiments can be key in providing the necessary spacelike separation.

A simple setup to test quantum entanglement for photon pairs is shown in Figure 2. A laser is sent through an opticalcrystal, which creates pairs of photons possessing an entangled linear polarization state. Each photon is sent through apolarizer, oriented at angles and , and then detected at D1 and D2 respectively. By tuning the orientations of thepolarizers, it is possible to test the quantum correlations and observe if quantum entanglement violates local realism.(This correlation can also be utilized for quantum cryptography.)

In the case of ultra-long-range Bell tests, two alternative scenarios may be considered. The rst is a symmetric setup,in which the quantum source is placed equidistant from two receivers. This is conceptually the most simple approach.A logistically easier (and therefore cheaper) alternative is to perform the experiment with one receiver, Bob, located atsome distance, and a second receiver, Alice, at the same site as the entangled photon source. For example, this mightrepresent satellite and ground receivers, respectively. Within this second scenario, however, if Alices measurement ismade on her photon immediately as it emerges from the source, then the two measurement events will not be spacelikeseparated, and the results from the experiment will be subject to locality and freedom-of-choice loopholes [ 24].

To close these loopholes, one can insert a delay before Alices measurement, such as a length of bre-optic cable ora quantum memory device. The appropriate choice of such a delay makes the measurement events spacelike separated,


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CONTENTS 8

Crystal

D1

D2

| BellLaser AND

CG

CounterN( , )

Figure 2. Schematic view of a simple Bell-test experiment with entangled photons. Entangled pairs of photons arecreated in parametric down-conversion of a laser passing through an optical crystal. The entanglement properties of thedetected photons are measured in two analyzers. Time-correlation of the photon detection signals is used to identifydetections arising from photons that were generated as pairs, typically via a logical AND gate. and represent twopossible measurement settings for each detector, and N (, ) the counts corresponding to each pair of settings.

giving an experimental setup in which both the locality and freedom-of-choice loopholes can be closed while requiringonly one measurement to take place at a distant site [24].

space

time

E g

X 1

E s

t 1

X 2

E g

E s

Figure 3. Effective distance between receivers in an asymmetric Bell experiment. Left: spacetime diagram of theexperiment in Alices frame, with two measurement events E g and E s on the ground (Alice) and on a satellite (Bob)

respectively. Right: spacetime diagram from perspective of reference frame in which the measurement events E g and E sare simultaneous. The effective spatial separation of the measurement events X 2 is much smaller than the altitude of theorbiting satellite. In each case the green arrow indicates the velocity of the other frame.

Figure 3 illustrates such an asymmetric Bell test experiment, with a spacetime diagram drawn from the perspectiveof two reference frames. The gure on the left is drawn in Alices frame, the frame of a ground station in which theentangled photon source is located. The emission event is colored in blue, and the two reception events are indicatedby red dots, E g occurring at the ground station (Alice) a short time t1 after the emission event, and E s occurring lateron an orbiting satellite (Bob) at an altitude of X 1 above the ground station.

The gure on the right shows the same experimental scenario, from the perspective of a reference frame which ismoving at a velocity v, away from the Earths surface, such that the two detection events are simultaneous. Accordingto the right reference frame, the detection events occur at a spatial separation of X 2 . The Lorentz invariant distancebetween the two events is

( x)2 (c t)2 = X

1 2 [c(X 1 /c t1 )]2 = X 2 . (1)For a satellite orbiting at X 1 = 1000 km and a quantum memory device which provides a t1 = 20 s delay, thisgives an effective separation of X 2 = 109km. The two reference frames would be travelling at a relative speed of v = X 1 ct 1X 1 = 0 .994c, which corresponds to a boost factor = 83 .6. For comparison, the Bell test detailed in Ref. [24],for which the Earth-frame distance was 144km (and which employed 6 km of optical bre to yield a delay of 20 s) hada corresponding Lorentz invariant spearation of the detectors of 41 km.

2.3. Bell test with human observers

Leggett raises the concern that the locality loophole in Bell tests may not yet be completely closed experimentally [25].He asserts that


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A truly denitive blocking of this loophole would presumably require that the detection be directly conductedby two human observers with a spatial separation such that the signal transit time exceeds human reactiontimes, a few hundred milliseconds (i.e. a separation of several tens of thousand kilometers). Given theextraordinary progress made in quantum communication in recent years, this goal may not be indenitelyfar in the future.A Bell experiment employing human observers could be performed at the scale of the Moons orbit about the

Earth. For example, one observer may sit in a lab on the lunar surface, while another sits on the Earths surface, withan entangled photon source located in a satellite at high Earth orbit. The possibility of employing humans to decideon the measurement setting in a Bell test leads to interesting tests of the role of free-will in quantum mechanics, asmentioned in [26].

2.4. Bell test with detectors in relative motion

This type of experiment is of interest for its relevance to the foundations of quantum mechanics. As already discussed inSection 2.2, some interpretations of quantum mechanics impart ontological physical meaning to wavefunction collapse.For such interpretations, as discussed above, the concept of quantum state reduction can lead to problems when relativityis considered, in particular with situations in which spacelike separated measurements take place [ 20, 21]. Otherinterpretations, such as the ensemble interpretation [ 27] or consistent histories [ 28] will be perfectly compatible with theexpected outcomes of these experiments [29].

Space platforms allow for (and in some situations, mandate) relative motion between a pair of observers at differentlocations with considerably less restrictions than on Earth. By exploiting this, one is able to test interpretations of wavefunction collapse in scenarios for which the two observers disagree on the relative time ordering of the measurementevents. If the two distant observers measure the photons, events S 1 and S 2 , under a large relative velocity and spatialseparation, then due to special relativity their lines of instantaneity (isochronous planes) will become shifted by ameasurable amount. This is a relativistic generalization of Bohms version of the Einstein-Podolsky-Rosen paradox;see Ref. [30] (Section 6.1) and Figure 4. Due to the mutual time shifts, the two observers may each measure theirparticle later than the other (an after-after scenario) in the case that the observers move apart from each other, oreach measure their particle earlier than the other (earlier-earlier), in the case of approaching motion.

1

1x 2y

2

2

1

t

Test of Test of S S

t

z

z

Figure 4. From Ref. [ 30], this spacetime diagram shows the coordinate systems and the locations of the two tests, S 1and S 2 . The t 1 and t 2 axes are the world lines of observers who are receding from each other. In each Lorentz frame, thez1 and z2 axes are isochronous: t 1 = 0 and t 2 = 0, respectively.

One picture of entanglement has it that the rst measurement inuences the outcome of the secondmeasurement, perhaps non-locally. However, this picture breaks down in a situation in which special relativity mandatesthat the time-ordering of events is ambiguous. The probabilities predicted by quantum theory do not depend on thetime-ordering of spacelike events, so its predictions will not be changed. Yet this timing paradox leaves us with somepuzzles about understanding the physical reality of quantum states and the non-local collapse of the wave functions.Essentially, in such an experiment we cannot even objectively dene the non-local update of one system because of themeasurements performed on the other. It is important to note that it is the measurements external interventions [29]into the system and not entanglement that are responsible for the paradox. If a pair of spin-half particles isprepared in a direct product state and the local measurements on the particles are mutually spacelike, then the state


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CONTENTS 10

of the system does not have a Lorentz-covariant transformation law outside the common past and the common futureof these interventions [31]. The Liouville function of two classical particles that are subject to two mutually spacelikestochastic events has the same property [31].

With space platforms it is possible to reach relative velocities and spatial separations which allow this paradoxicalsituation to be observed and explored. The respective time shift between the two observers by the Lorentz transformationis

t =1

1 v20 /c 2 t

v0 xc2 , (2)

and by setting the origin of reference frame t to be zero, we obtain:

t =v0 xc2

. (3)

As an example, by inserting the velocity of 2 7.5 km/s = 15km/s for the relative motion of two LEO satellites, weobtain the following temporal shift per kilometer:t/x =

v0c2

=15km/ s

(3 105 km/ s)2= 166 ps/km (4)

Assuming fast random number generators and optical switches allow randomly setting the analyzers every 10ns (whichis more than 10 times faster than reported in [24]), this would require a minimal separation of the two satellite observersof about 60 km. Assuming the measurements can be performed up to the maximal separation for two satellites in LEO

of about 1500 km, the time available for measurements would be about one and a half minutes, which should be sufficientfor a valid Bell test.

Such an experiment comes with another important requirement, namely that the emission of photon pairs from thesource must be very well timed, so that the two photons travel exactly the same time from the source to their respectivemeasurement stations. The synchronicity requirement is better than 10 ns for the shortest separation of 60 km, andexpands up to 250 ns at 1500km, which is challenging but seems achievable.

2.5. Bell experiments with macroscopic amplication

At rst sight, the quantum projection postulate tells us that something alters when a measurement takes place: thestate vector of the measured quantum system changes discontinuously as a result of the measurement. Of course, oneview of quantum theory, in line with several different interpretational ideas, is that this collapse of the wave functionis merely a mathematical operation that physicists carry out, and does not represent any real physical process. While

that view may turn out to be correct, it presently leaves us with some fundamental puzzles. If we cannot extract adescription of physical reality from the quantum wave function, then how do we nd a description of physical realitywithin quantum theory? Or if we cant, how do we make sense of a fundamental physical theory that doesnt describean objective reality (and so doesnt seem to describe observers or measuring devices either)? But if we can extract adescription of physical reality from the wave function, why wouldnt we expect that reality to change discontinuouslywhen the wave function does?

On the other hand, the idea that wave function collapse might be a real physical process also runs into immediatedifficulties, one of which is that, as normally formulated, the description of wave function collapse depends on oneschoice of reference frame, and so taking it to be real seems to conict with special relativity.

Reviewing all the arguments and counter-arguments and the different interpretational ideas supporting each isbeyond our scope here. We simply want to note an interesting non-standard line of thought about wave functioncollapse that motivates new experimental tests of quantum theory.

To motivate this, we rst suppose that wave function collapse is a real objectively dened process, and moreoverone that is localized : collapses occur as the result of events that take place at denite spacetime points. On this view,collapses are not fundamentally dened in terms of observers or measurements. The collapse process obeys some as yetunknown mathematical laws perhaps, for example, some relativistic generalization of the Ghirardi-Rimini-Weber-Pearle (GRWP) [32, 33] spontaneous collapse models from which the usual account of collapses taking place as theresult of measurements in our experiments emerges as a special case and an approximation.

We also suppose that collapses propagate in a way that respects special relativity. A collapse event at a spacetimepoint P only affects physics in the future light-cone of P : in particular, it does not affect the probabilities of any othercollapse events at points spacelike separated from P .

Taken together, these hypotheses give a way of dening an alternative to standard quantum theory, which wecall causal quantum theory , that makes very different predictions from standard quantum theory [22, 23]. To denecausal quantum theory precisely, we would need a precise formulation of the dynamical collapse laws, which needs a


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CONTENTS 11

relativistic collapse theory. (So causal quantum theory is really an umbrella term for a class of theories.) But we can getsome intuition about the nature of the alternative by supposing that the existing (non-relativistic) GRWP models are agood approximation to this unknown relativistic theory for experiments where relativistic quantum effects are relativelynegligible. In particular, we can look at the implications for Bell experiments.

Causal quantum theory predicts that, in a Bell test in which measurements on two separated particles in a singletstate are genuinely completed (i.e. the relevant underlying collapse events take place) in spacelike separated regions,essentially no correlations will be observed in the measurement outcomes. Observing spin up on one wing does notaffect the probabilities of observing spin up or down on the other. This is, of course, very different from the correlationspredicted by standard quantum theory and apparently observed in essentially all Bell experiments to date. It leads topredictions of outcomes that are very unlikely, according to standard quantum theory. These predictions nonethelessnever actually cause a logical contradiction: causal quantum theory is self-consistent [22, 23].

At rst sight, causal quantum theory may seem to be clearly refuted by the overwhelming experimental evidence forBell correlations. However, this in fact raises a new worry about all existing Bell tests: (how) do we know that collapseevents sufficient to effectively dene an irreversible measurement do in fact take place in spacelike separated regions inthe two wings of the experiment? Is there a possible alternative, namely that the wave function remains uncollapseduntil a later point in the measurement chain, when the signals produced in the two wings are brought together andcompared? If so, causal quantum theory and standard quantum theory would then both predict the standard quantumcorrelations actually observed. But then no Bell experiment to date would actually have succeeded in demonstratingquantum non-locality. After all, causal quantum theory is, as its name suggests, a locally causal theory. If it is consistentwith the results of any given experiment, then by denition this experiment cannot have demonstrated non-locality innature.

Quantitative estimates suggest this is indeed a live issue, at least in principle. With an appropriate choice of collapseparameters, GRWP models would indeed predict that there are essentially no collapses in spacelike separated regions onthe two wings of standard Bell test experiments. For example, neither the avalanches of photons generated by detectinga photon in a photo-multiplier, nor the electrical currents that ensue, are sufficiently macroscopic to (necessarily)correspond to a GRWP collapse in anything like a sufficiently short time. There is a loophole the collapse locality loophole in all Bell experiments to date. One can nd collapse models, and so versions of causal quantum theory, inwhich none of these experiments would (essentially) ever cause spacelike separated collapse events in the two wings. Nocollapse occurs, and so no denite outcomes or correlations are generated, until after the signals are brought togetherand compared. If this description were correct, quantum non-locality would never actually have been demonstrated.

To close this loophole, one needs to carry out Bell tests in which the measurements in the two wings producemacroscopically distinguishable outcomes in spacelike separated regions. In principle, one simple way to do this, in line

with the GRWP model and also with Penrose [ 34] and Diosis [35] intuitions regarding a link between gravitation andcollapse, is to arrange experiments so that massive objects are quickly moved to different positions depending on themeasurement outcomes. The technological challenge of arranging for a mass to move as the result of a measurementover timescales short compared to terrestrially achievable Bell experiment separations is, however, considerable.

Nonetheless, Salart et al. [ 36] were able to carry out a beautiful terrestrial experiment, motivated by these ideas,exploiting the rapid deformation of piezocrystals when a signal voltage is applied. In their experiment, a gold-surfacedmirror measuring 3 2 0.15 mm and weighing 2 mg is displaced by a distance of at least 12 .6 nm by a piezocrystalactivated by a detector in a Bell experiment. The displacement is complete within 6 .1 s after the photon enters thedetector. Using estimates due to Penrose [34] and Diosi [35] for a plausible collapse time if the collapse is related to thegravitational energy of the mass distribution associated with the two superposed mirror states, Salart et al. obtained acollapse time of 1 s, suggesting that the measurement is complete and a collapse has taken place within 7.1 s. Thetwo wings of their experiment were separated by 18 km, i.e. c 60 s, so that on this interpretation the collapses in thetwo wings are spacelike separated. As their results agreed with standard quantum theory, they refute causal quantumtheory for a range of collapse model parameters.That said, neither Penrose nor Diosis estimates derive from a consistent theory of gravity-related collapse. Thereare a variety of ways to try to build such models (see, e.g., Refs. [37] and [38]). The predicted collapse rates in anygiven experiment are very sensitive to details of such models, which are not currently xed by any compelling theoreticalprinciple. Moreover, the earlier GRWP collapse models [32, 33] (which do not link collapses directly with gravity) andother types of collapse model are also well-motivated. For these models too, the predicted collapse rates in any givenexperiment are very sensitive to uncertain details and ad-hoc choices. For example, one possible extrapolation [39] of the original GRW collapse model to indistinguishable particles would suggest a collapse time of > 102 s for the mirrorsuperposition states in the Salart et al. experiment.

The Salart et al. experiment has thus by no means closed the collapse locality loophole completely. To do that,we would need to arrange a Bell experiment in which different measurement outcomes produce matter distributions


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CONTENTS 12

so macroscopically distinct, and in which the separation between the wings is long enough in natural units, that noremotely plausible collapse model can predict that the systems on the two wings remain in superposition throughoutthe experiment.

This can and presumably ultimately will be achieved by Bell experiments based in deep space, with very largeseparations between the wings. Interestingly, though, even near-Earth experiments can achieve a great deal. A Bellexperiment on a scale of 10 3 105 km would be able to exclude any model that predicts collapse times shorter than 3300 ms. The upper end of this range compares well with human reaction times (

100 ms); even the lower end compares

well to the time required to create an effect on a scale of meters at explosive detonation velocities ( 103 104 m/s). Tohave any plausible motivation, a collapse model must produce collapses of macroscopic superpositions perceptible byhumans on timescales short compared to those we can discriminate. A model which fails to do this has to explain whywe see denite outcomes while being, at least for some perceptible length of time, actually in indenite superpositions.

Some creative engineering ingenuity is required to devise macroscopic amplications well suited for near-Earth Bellexperiments, given the constraints of practicality and repeatability. Mechanical amplications of a detector signal tomove macroscopic masses are presumably more practical and more efficient than those relying on human intervention,and more repeatable than those involving explosives. For the moment, we offer this design problem as a challenge toexperimental colleagues, encouraged by the above gures, which suggest that even near-Earth, well-designed experimentsshould be able to close the collapse locality loophole very signicantly, and perhaps indeed completely.

2.6. Bimetric gravity

Moffat has questioned whether quantum entanglement is divorced from our common intuitive ideas about spacetime andcausality, and proposed a relativistically causal description of quantum entanglement in terms of a bimetric theory of gravity [40]. Such a theory should be testable with a Bell experiment in which the two receivers are moving relativisticallywith respect to each other, as described in Section 2.4.

Quantum entanglement, according to the standard interpretation, is a purely quantum phenomenon and classicalconcepts associated with causally connected events in spacetime are absent. This is the point of view promulgated bystandard quantum mechanics. One should abandon any notion that physical space plays a signicant role for distantcorrelations of degrees of freedom associated with particles in entangled quantum states. For those who remain troubledby this abandonment of a spatial connection between entangled states, it is not clear how attempting to change quantummechanics would help matters. This leaves the possibility that classical special relativity is too restrictive to allow for acomplete spacetime description of quantum entanglement. An alternative scenario is based on a bimetric descriptionof spacetime. In a bimetric theory, the light cones of two metrics describe classical and quantum spacetimes [40].The quantum spacetime is triggered by a measurement of a quantum system and allows the propagation of quantuminformation at superluminal speeds prohibited by the lightcone in the classical spacetime.

The satellite experiment proposed in Section 2.2 may already be sufficient to provide evidence for bimetric scenarios.For example, in the case that the quantum mechanical metric of Ref. [ 40] has a Lorentzian signature (as is depictedin Figure 1 of Ref. [40]) then the two measurement events at E g and E s of Figure 3 must be causally related by thequantum metric if quantum correlations are observed. It is possible that by changing the delay on the ground, or bya changing distance to a satellite in an eccentric orbit, the measurement events may cease to be in causal contact,according to the quantum metric, in which case the quantum correlations predicted by the theory would vanish.

3. Relativistic effects in quantum information theory

With the possible exception of Section 5, spacetime in our discussion is either (weakly) curved or an (approximately) atxed background for propagation of polarization qubits. The simplest approximation that is tacitly assumed in most

of the discussions of long-distance quantum information processing is to treat photons as massless point particles thatmove on the rays prescribed by geometric optics (e.g., null geodesics in vacuo) and carry transversal polarizations. Thelatter can be described either in terms of a two-dimensional Hilbert space or a complex three-dimensional vector, whichis orthogonal to photons momentum [ 41, 42] (Section 3.1.1). Evolution of a polarization along the ray is described bythe rst post-eikonal approximation [ 43], and the resulting rotation is interpreted as quantum phase. Validity of thisapproximation is justied because of the scales involved. The nite-width wave packet effects, while introducing manyinteresting features, are of higher order [ 41, 44].

Therefore, photons follow null trajectories in spacetime with tangent four-vectors k, k2 = 0, and the spacelikepolarization vectors f , f 2 = 1, are parallel-transported along the rays [ 45, 46],

k k = 0 , k f = 0 , (5)


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CONTENTS 13

where k is a covariant derivative along k.All experiments which contain at least one receiver in orbit will involve high-precision reference frame alignment,

in a general relativistic setting. This raises some interesting issues, as discussed below.

3.1. Special relativistic effects

3.1.1. Lorentz transformations and polarization Every Lorentz transformation that connects two reference frames

results in a unitary operator that connects the two descriptions of a quantum state, | = U ()| [47]. The unitariesU () are obtained using Wigners induced representation of the Poincare group [48, 47]. The rst step in its constructionis also used to build a polarization basis in a stationary curved spacetime (Section 3.2.1).

Single-particle states belong to some irreducible representation. Two invariants the mass m and the intrinsicspin j label the representation. The basis states are labelled by three components of the 4-momentum p = ( p0 , p )and the spin along a particular direction. Hence a generic state is given by

| = d( p) ( p)| p, , (6)

where d( p) is the Lorentz-invariant measure and the momentum and spin eigenstates are -normalized and are completeon the one-particle Hilbert space.

The single-photon states are labelled by momentum p and helicity p = 1, so the state with a denite momentumis given by = 1 | p, p , where

| +

|2 +

|

|2 = 1. Polarization states are also labelled by 3-vectors p , p

p = 0,

that correspond to the two senses of polarization of classical electromagnetic waves. An alternative labelling of the samestate, therefore, is = 1 | p, p [49, 42].Action of the unitary operator U () is represented in Figure 5. It is derived with respect to the standard 4-momentum, which for photons is taken to be kR = (1 , 0, 0, 1), and the standard Lorentz transformation

L(k) = R(k )B z (u), (7)

where B z (u) is a pure boost along the z-axis with a velocity u that takes kR to ( |k |, 0, 0, |k |) and R(k ) is the standardrotation that carries the z-axis into the direction of the unit vector k [47]. If k has polar and azimuthal angles and ,the standard rotation R(k ) is accomplished by a rotation by around the y-axis, followed by a rotation by aroundthe z-axis.

Explicitly, the transformation is given by

U ()

| p, =

D [W (, p)]

| p, , (8)

where D are the matrix elements of the representation of the Wigner little group element W (, p) L 1 ( p)L( p)that leaves kR invariant, kR = WkR .spin

momentum

classical info

D

Figure 5. Relativistic state transformation as a quantum circuit: the gate D which represents the matrix D [W ( , p )]is controlled by both the classical information and the momentum p, which is itself subject to the classical information(i.e. the Lorentz transformation that relates the reference frames).

Helicity is invariant under proper Lorentz transformation, but the basis states acquire phases. An arbitrary littlegroup element for a massless particle is decomposed according to W (, p) = S (, )Rz ( ), where the elements S (, )do not correspond to the physical degrees of freedom [47]. As a result, the little group elements are represented by

D = exp( i ) , = 1. (9)For pure rotations, = R (in this case the same letter is used for both a 4D matrix and its non-trivial 3D block),

the phase has a particularly simple form [ 41, 44]. We decompose an arbitrary rotation R , k = R p , as

R = R k ( )R(k )R 1 (p ), (10)


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CONTENTS 14

where R k ( ) characterizes a rotation around k , and R(k ) and R(p ) are the standard rotations that carry the z-axis tok and p , respectively. Then

W (R , p) = R z ( ). (11)

A practical description of polarization states is given by spatial vectors that correspond to the classical polarizationdirections. Taking again kR as the reference momentum, two basis vectors of linear polarization are 1k R = (1 , 0, 0) and

2k R = (0 , 1, 0), while to the right and left circular polarizations correspond

k R = (

1k R

i 2k R )/ 2.

Phases of the states obtained by the standard Lorentz transformations L(k) are set to 1. Since the standard boostB z (u) leaves the four-vector (0 , k R ) invariant, we dene a polarization basis for any k as

k R(k ) k R | k, L(k)|kR , , (12)

while the transformation of polarization vectors under an arbitrary R is given by the rotation itself. Indeed,R (p ) = R k ( ) (k ) U (R )( + | p, + + | p, ) = + ei |k, + + e i |k, . (13)

For a general Lorentz transformations the triad ( 1p , 2p , p ) is still rigidly rotated, but in a more complicatedfashion [42, 44, 50].

Since LEO satellite velocities are of the order v/c 10 5 the rst order expansion for a pure boost [ 50] is sufficientto estimate the induced phase. If the photon propagates in the rst frame along p = (sin cos , sin sin , cos ) (weassume < / 2) and the frames are related by the boost v(sin b cos b, sin b sin b , cos b), then

= 12 tan 12 sin b sin( b) vc . (14)The effect is most pronounced ( 12 v/c 10 5 ) when the light is sent in a direction perpendicular to thereference z-axis, with the receiver velocity being perpendicular to both.The Lorentz transformation also inuences the diffraction angle for wave packets [41]. For a detector moving with

respect to the emitter with velocity v along the propagation direction of a narrow beam ( v/c ), the diffraction anglechanges to

= 1 + v/c1 v/c . (15)3.1.2. Entanglement As long as nite wave-packet width effects can be ignored [ 41] a bipartite or multipartitepolarization entanglement is preserved. This can be seen as follows. Lorentz boosts do not create spin-momentumentanglement when acting on eigenstates of momentum, and the effect of a boost on a pair is implemented on bothparticles by local unitary transformations, which are known to preserve entanglement. This conclusion is valid for bothmassive and massless particles. Since the momenta are known precisely, the phases that polarization states acquire areknown unambiguously, and in principle can be reversed. Hence, for a maximally entangled pair in the laboratory frame,the directions of perfect correlation for two photons still exist in any frame, even if they are different from the laboratorydirections [ 51, 52].

3.2. General relativistic effects

3.2.1. Relativistic frame dragging A somewhat imprecise term frame dragging [53, 54, 55, 56] refers to a numberof phenomena that, in a eld of a slowly rotating mass, can be attributed to the presence of the mixed spacetimecomponents of the metric h i = g 0 i [45, 53]. In contrast to geodetic effects that result from mass-energy density, theformer are induced by the mass-energy currents of the source. The eld h i plays a role analogous to the vector potentialof electromagnetism. We refer to these and similar phenomena as gravitomagnetic [53]. A frame-independent indicatorof gravitomagnetism is a non-zero value of the pseudo-invariant R R (where R is a dual of the Riemanntensor), which for an isolated system is proportional to its angular momentum J [53].

Two of the best-known gravitomagnetic effects Lense-Thirring precession of the orbit of a test particle and Schiff precession of the axis of a gyroscope were used in precision tests of general relativity [57, 58, 59]. In the near-Earth environment the effects are small and challenging to measure: gravitomagnetic precession rates for the LAGEOSexperiment (Lense-Thirring effect) and the Gravity Probe-B (Schiff precession) are 31 arc msec/yr and 39 arc msec/yr,respectively.

Gravitomagnetism also inuences propagation and polarization of electromagnetic waves. Changes in polarization(a gravitomagnetic/Faraday/Rytov-Skrotski rotation [60]) are operationally meaningful only with respect to a localpolarization basis [ 61].


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CONTENTS 15

A reference-frame term [ 61, 62] contributes to the polarization rotation alongside the Machian gravitomagneticeffect [63, 64]. In a stationary spacetime the Landau-Lifshitz 1+3 formalism [ 65] lets us rewrite the 4D parallel transportequations in a 3D form [64], demonstrating the joint rotation of the unit polarization f and the unit wave vector k [62],

D kd

= k ,D f d

= f , (16)where D/d is a 3D covariant derivative (with respect to an affine parameter ). The angular velocity of rotation is

given by = 2 ( k )k E g k . (17)

Here the gravitoelectric term E g reduces to the Newtonian gravitational acceleration in a non-relativistic limit, and = B g /2 is a gravitomagnetic term.

Transversality of electromagnetic waves determines the plane of polarization for each propagation direction, buta choice of a polarization basis is still free. Only by referring to these bases can the evolution of f be interpreted aspolarization rotation. In Minkowski spacetime this choice is made once for all spacetime points, as in Section 3.1.1.Comparisons between different points on a curved background require connections.

Consider some choice of a (linear) polarization basis b 1 , 2 (k ) along a photons path. By setting f = b 1 at the startingpoint of the trajectory, the phase appears as f () = cos b 1 + sin b 2 , so

dd

=1

f

b 1

D f d b 2 + f

D b 2d

= k +1

f

b 1

f D b 2d

. (18)

The term k is the Machian contribution [ 63], but the reference-frame term may be equally or even more important [61].A commonly held view that no polarization phase is accrued in the Schwarzschild spacetime can be substantiatedin the so-called Newton gauge [61], where the ducial z-axis (Section 3.1.1) is chosen along the free-fall accelerationas seen by a static observer, and the standard polarization direction b 2 is directed along z k . In any spacetime, aphase accumulated along a closed path is gauge-independent and can be expressed as an area integral of an appropriatecurvature [ 62].

Two examples (in the Newton gauge) will illustrate the order of magnitude of the effects. In the Kerr spacetime,the calculations are simplied by having a sufficient number of conserved quantities [ 46, 65]. Outgoing geodesics in theprincipal null congruence satisfy a number of remarkable properties (including that these are trajectories of a constantspherical angle ) and can be easily integrated [46]. The leading order of polarization rotation is found by a directintegration,

sin = J

Mc1r 1

1r 2 cos , (19)

where r 1 and r 2 are initial and nal radial coordinates, respectively. The Earths angular momentum is J = 5 .861040cm2 g/s and its mass is M = 5 .98 1027 g. Sending a photon on such a trajectory, starting at r 1 = 12,270 km (thesemimajor axis of the LAGEOS satellite orbit [58]), at the latitude of 45 and taking r 2 , we obtain the rotationof 39 arcmsec in a single run, and nearly twice this amount if the light is sent from the ground. This result,however, depends on the special initial conditions [61].

The results typically scale as the inverse square of a typical minimal distance, r 2typical . For example [ 61, 62]a photon that was emitted and detected far from the spinning gravitating body, but passed close to it. In a special caseof emission along the axis of rotation, we get in the leading order

sin =4GJ s2 c3

, (20)

where s is the impact parameter [62]. The antiparallel initial direction gives the opposite sign. For the Earth (and the

impact parameter being its radius), the resulting phase is minuscule: 3 10 7

arcmsec.The above examples provide an estimate for the order of magnitude of a potentially observable effect, however theyare not gauge invariant. The latter requires the photons to traverse a closed path. While such an experiment does notrequire a complicated alignment of the polarizers, at least three nodes are necessary to produce a path that enclosesa non-zero area. The conceptually simplest path is formed in the following setting. The trajectory starts parallel tothe axis of rotation with an impact parameter s1 (and the initial angle 1 = ). Far from the gravitating body (so itsinuence can be ignored), the outgoing photon is twice reected and sent in again with the impact parameter s2 andthe initial angle 2 = 0. After the second scattering and appropriate reections it is returned to the initial position withthe initial value of the momentum. Then

=4GJ

c31s21

1s22

. (21)


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CONTENTS 16

3.2.2. Entanglement in the presence of curvature Consider creating an entangled pair, say of polarized photons ormassive spins, which are separated so far that curvature and the nontriviality of parallel transport becomes signicant.When considering an EPRB-type experiment on two such separated spins, the question arises as to what directionthe reference frame in the neighborhood of one particle corresponds to in the neighborhood of another. Assumingthere is curvature, parallel transport of the z-direction from one spinning particle to the other is ambiguous since itis path dependent. Even though there may be a unique geodesic between the two EPRB measurements, the commonassumption is that one needs to calculate the parallel transport of one spins z-axis back (in time) along its worldlineto where the pair was created and then parallel transport that direction forward along the other spins worldline to theother spin. On one hand, if the curvature is known, this should allow one to predict how the reference frames of theEPRB measurements ought to be aligned to meet the EPRB predictions. Experimental verication would constitutethe measurement of an effect that involves essential parts of gravity and quantum theory, namely both curvature andentanglement.

Similarly to the special-relativistic effects of Section 3.1.2, any gravitomagnetic phase (c.f. Section 3.2.1)which arises along the path of particles can be considered as arising from a local gate, and thus does not changethe entanglement. So long as the approximation of point particles that move on well-dened trajectories holds, theparallel transport of the reference directions (for massive particles) or local Newton gauge construction for photons willallow preservation of the entanglement.

On the other hand, if such an experiment could be set up, it would allow one also to use a quantum effect to measureaspects of curvature. To this end, one would perform an EPRB-type experiment with such entangled pairs, initially notknowing the curvature along the paths that the entangled particles took on their way from where they were generatedto where they were measured. The aim would be to nd and record that relative alignment of the local reference frameswhich optimizes the EPRB effect, i.e. which corresponds to the correct identication of the local reference frames. Thiswould constitute a curvature measurement in the following sense. Assume that the curvature between the source of entangled particles and the EPRB measurement apparatuses changes. This could be detected in the EPRB experimentson subsequently produced entangled pairs, because for them the optimal alignment of the local z axes would be different.

While the effects are tiny, and perhaps classical optical experiments may have a greater chance of success inmeasuring properties of the gravitational eld, using entangled states to probe spacetime geometry is interesting at leastas a question of principle. This proposal is also discussed in Ref. [13]. There the authors propose using NOON states inan interferometer to enhance the observed signal.

In principle, EPRB experiments on entangled pairs of spins that traveled through regions of signicant curvatureneed not only combine curvature with entanglement. They could also combine curvature and entanglement with quantumdelocalization. This could be achieved if the entangled spins are made each to travel in a wave packet that is as large

as the curvature scale. In this case, there is no unique path that the entangled particles take and therefore the rotationof their spins that arises from the curvature is quantum uncertain. Presumably, this still leads to an effectively xedorientation between the two z axes so that the full EPRB effect is observed. That is because as long as the particles donot signicantly impact the spacetime, spacetime is not observing and not decohering the particles paths.

This setup is likely to be exceedingly difficult to implement experimentally because of the difficultly of keeping trackof individual entangled pairs over scales that are large enough so that the gravitational nontriviality of parallel transportbecomes signicant. However, intriguingly, this setup fundamentally involves both quantum and gravity effects withoutrequiring high energies, much less energies close to the Planck scale. What makes this possible is of course that we arehere not yet looking for quantum effects of gravity but merely for quantum effects that are inuenced by gravity.

3.3. The Fermi problem and spacelike entanglement tests

The Fermi problem [ 66] is a gedanken experiment put forward by Fermi as a causality check at a quantum level. The

experiment consists of two spacelike separated atoms A and B at relative rest separated by a distance R. At somecommon proper time t = 0, A is in an excited state while B is in the ground state, and the electromagnetic eld isprepared in the vacuum state. The question that Fermi asked is Can the atom A decay to the ground state and provokethe excitation of the atom B at a time t < R/c ?

Due to intrinsic characteristics of quantum electrodynamics (namely the non-vanishing propagator of theelectromagnetic interaction outside the light cone) the answer to this question has provoked a lengthy controversy [67,68, 69] on the possible causal behaviour of the probability of excitation of the atom B.

Recently, a series of works demonstrated that a proper approach to this problem requires a reformulation in termsof non-signaling and quantum non-locality [ 70]. While the atom B can denitely be excited outside the light cone, theconditioned probability of B being excited after A has been measured in the ground state is strictly causal.


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CONTENTS 17

Although some experiments to test this quantum foundational question have been proposed in platforms such assuperconducting circuits and optical cavities, none of them has yet been carried out.

An experimental test of the Fermi problem would constitute a fundamental test of quantum eld theory. Theexperimental difficulties of such an experiment come from the initial state preparation and the control of the processof switching on and off the effective interactions. Being able to satisfactorily acknowledge this result in an experimentwould imply that we can guarantee that the detectors remain causally disconnected during the experiment while theyare coupled to the same quantum eld. Let us call this difficulty the loophole of spacelike detection.

The Fermi problem is strongly related with the phenomenon rst reported by Retzker, Reznik and Silman [71, 72]that two spacelike separated detectors can extract entanglement from the vacuum state of the eld. In this scenariowe have two spacelike separated detectors in the ground state at relative rest in a at spacetime. These detectors arecoupled to a massless eld in a time dependent way that, prior to the activation of the interaction, is in the vacuumstate.

At some point the interaction is switched on and the detectors interact with the eld during the short timethey remain causally disconnected, then the interaction is switched off. The result is that the nal state of thedetectors presents quantum entanglement. Depending on the arrangement of spacelike separated detectors the extractedentanglement can be very strong, even nearly reaching the maximally entangled state for the two detectors [73].

The study of this problem connects with the fundamental results found in algebraic quantum eld theory aboutthe correlations in the vacuum state of the eld between local algebras of observables living in spacelike separatedpatches of the spacetime [74, 75]. While detection of this phenomenon would be a test of the quantum non-locality inquantum eld theory, and some proposals have been sketched [72], an experimental test is still to appear.

The amount of generated entanglement between two detectors at rest separated by a distance R decays exponentiallywith the ratio R/ (cT ) where T is the interaction time. Namely, a lower bound for the entanglement of the two detectorssubsystem after the interaction time T is [71]

N exp R

cT

3

where N is the negativity of the two detectors partial state [76]. For the detection of spacelike entanglement in alaboratory experiment, we again run into problems with the spacelike detection loophole: it is difficult to keep therelevant interaction switched on only during the time that the detectors are spacelike separated.

Indeed, a carefully controlled laboratory setting where we can have complete control on the interaction would haveserious difficulties in carrying out the state preparation and state readout of the detectors within the time interval (of order of nanoseconds) that the detectors remain spacelike separated.

This spacelike separation loophole can be overcome using non Earth-bound experimental platforms. The longerthe distance between the two detectors, the longer we can keep the condition of spacelike separation. That meansincreasing the interaction time and therefore dramatically reducing the problems with the interaction switching anddetector readout (see Table 2).

To ensure that we would be able to detect this entanglement extraction, the detectors have to couple to the sameeld mode and coherence must be kept in the quantum eld mode probed. That means that any source of noise inthe relevant bandwidth should be kept to a minimum. The relevant bandwidth is given by the spectral response of thedetectors. For instance, if the detectors are atom-based the spectral response of the detector is given by the gap of theatomic transition used.

As discussed in [71], under these conditions, enough entanglement to violate CHSH inequalities can be effectivelyextracted for any separation distance R. The amount of entanglement that can be achieved between the detectorsincreases with the frequency gap, so an experiment using atoms whose energy difference between the ground state andrst excitation is of the order of visible light will give better results than an experiment using detectors tuned to themicrowave or radio spectrum. This is benecial for a possible experimental implementation since isolating the modeprobed by the detectors from noise in the light spectrum is much easier than in the microwave or radio spectrum. To thisregard, the requirement to reproduce the exact theoretical scenario in [ 71] is that the eld mode probed is approximatelyin the vacuum state.

When going to longer separation distances we have to be very careful about this assumption on the mode with whichthe detectors interact: if the mode probed in this experiment is subjected to any kind of noise that makes invalid theassumption that the state of the eld is approximately the vacuum, then the experiment will deviate from the hypothesisof the published theoretical works mentioned above. However, achieving noise insulation of a quantum channel in spacedoes not seem extremely difficult. In absence of direct solar illumination, the most important source of noise, the CMB,can be avoided using visible light photons, which are widely detuned from the peak of the CMB spectrum (a thermalspectrum at T 3 K). Of course that would imply that the experiment should be screened from solar radiation. Provided


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CONTENTS 18

Kind of experiment Typical distances RSpace-like separationtimes R/c T

Feasibility

Tabletop 1 m 3 ns Earthbound 10km 30 s Possibly with superconduct-ing qubits [77]LEO satellite-based 1000 km 3ms Barely feasible with ions [ 78]GEO satellite-based

36,000km

0.1s Well feasible with ions/atoms

Moon-Earth 380,000 km 1s Feasible with macroscopic de-tectorsSolar System 1a.u. 500 s Well feasible with macro-scopic detectors

Table 2. Interaction times for detectors at xed relative distance in which the detectors only interact with the eldwhile they remain spacelike separated.

that the satellites are not receiving direct solar illumination, the noise in the visible light channel can be, at least inprinciple, kept to a minimum. This suggests that the experiment could benet from solar shielding of the spacelikeseparated detectors, with a design similar to what is planned for the James Webb Space Telescope [79], or strategicalplacement in the shadow of planets and moons, or maybe even on a hypothetical Lunar station.

The accomplishment of this kind of experiment would also have further implications on tests of quantum eld theoryin non-inertial frames and in the use of entanglement as a tool in cosmology, as we discuss in Section 4.3.

3.4. COW experiments

In 1975 Colella, Overhauser, and Werner (COW) performed an experiment which observed gravitational phase shift in aneutron beam interferometer [80]. The underlying motivation for this experiment (repeated with increasing precision overseveral years [81, 82]) is to test the classical principle of equivalence in the quantum limit. Despite many improvementsto the original experiment, a small discrepancy of 0.60.8% between theory and experiment remains [ 26].

For a beam of neutrons of mass m and wavelength entering an interferometer, the phase shift between the twosub-beams is

= m2 g2 2

A sin = 2g

v 2A sin

where A is the area enclosed by the trajectories of the two sub-beams, is the tilt angle of the interferometer above thehorizontal plane and g is the acceleration due to gravity. The second equality follows from the de Broglie wavelengthformula (for neutron velocity v) and in this sense the COW experiment can be regarded as a gravitational redshiftexperiment for massive particles. Upon replacing v c, this formula becomes identical to the phase shift formula forphotons along different paths in a constant gravitational eld [83].

It is therefore interesting to consider an optical version of the COW experiment using transmission of a coherentbeam to a satellite in LEO see Figure 6. At the Earths surface and in the moving satellite the beam is coherentlysplit by a semitransparent mirror, with one path going through l = 6 km of optical bre delay, while the other path isdirectly transmitted. The paths are recombined at the end to construct an interferometer. Since the upper bre sitsat a much higher gravitational potential than the lower, it will experience a different phase shift from the one on theground, which will be picked up by the interferometer.

Such an optical COW experiment could provide tests of gravitational redshift in the context of a quantum opticsexperiment. Specically, in a weak gravitational eld the redshift is given by the difference of the Newtonian gravitationalpotentials on the ground and at the satellites orbit,

=c2

ghc2

, (22)

where g is the free fall acceleration on the Earths surface and h is the altitude of the satellite. The resulting phasedifference between the two paths is = kl, where k is the wave number. Hence, taking the wavelength = 800nmand the altitude h 400km, one obtains quite a considerable phase difference:

=2l

ghc2

2rad . (23)

Furthermore, due to the large speed of the satellite (ca. 8 km/s) it will have moved by about 15 cm (depending onthe exact location of the satellite when the photons reach it) while the interferometer is closed, which effectively extends


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CONTENTS 19

Delay

BS

1

0

SPS

Delay

Ground

Satellite

v

Figure 6. Optical COW experiment. The two delays could each be up to 20 s using 6 km optical bres, during whicha LEO satellite will move about 15 centimeters. The two beams differ by an angle 31 arc msec.

the satellite based delay. This has two consequences, rstly that the interferometer paths now cover an actual areaand could be sensitive to rotational effects, and secondly, that in order to calibrate for zero-path-length, this delay, andconsequently the satellite speed, must be very accurately measured, which could provide useful information for analysisof satellite motion.

Note that this optical COW interferometry experiment would be carried out over much larger distances so that theEarths gravitational eld is not constant. The phase shift detected would then be general-relativistic as opposed to onedue to an accelerated frame (which the constant eld near the Earths surface is equivalent to). This would constitutethe rst direct measurement of quantum interference due to curved spacetime.

Higher order relativistic kinematic effects, similar to the Sagnac effect [54, 84, 85], especially for experiments witha pair of satellites, are also expected.

4. Tests of quantum eld theory in non-inertial frames

The tests in Section 3 consider relativistic effects on quantum mechanics and in special relativistic eld theory. Here wego further to explore how we might be able to probe the physics of quantum eld theory in accelerating frames. To ourknowledge, no direct experimental test of physics in this regime has ever been performed.

4.1. Test of the Unruh effect, entanglement delity and acceleration

An accelerated observer in the eld vacuum as dened by an inertial observer would experience a thermal bath in aphenomenon analogous to the Hawking effect in a black hole. This is called the Unruh effect [86]. For an accelerationa, the temperature of this thermal bath is

T = a

2cK , (24)

where K is the Boltzmann constant. At the surface of the Earth a 10m/s 2 , yielding T 4 10 20 K.The rst order effect of this thermalization is to act as a source of local noise in any mode of the eld that maybe probed by an accelerated detector. A more interesting effect is one in which (depending on the initial state of theeld) a system initially prepared in an entangled state in an inertial frame may experience a variation in its degree of entanglement when measured by receivers that are in relative acceleration. In the simplest scenario with a maximallyentangled state of the eld, it has been shown that entanglement degrades due to the presence of this Unruh noise [87, 88].

The magnitude of the acceleration necessary to observe these effects in, for example, an entangled state of theelectromagnetic eld [ 89] in a eld mode of frequency is

a c (25)which for a eld mode in the MHz spectrum would mean an acceleration of approximately a 1013 g. This is alreadymuch smaller that the acceleration a 1022 g that one needs in order to observe a thermal bath warmer than the CMB.However it is still too large to contemplate direct observation in the gravitational eld of any celestial body within the


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CONTENTS 20

solar system. A satellite-based experiment aiming to look for the resulting loss of entanglement between an acceleratingreceiver on the Earths surface and an inertial observer in orbit would be unable to see this effect.

The preceding results were obtained using what is known as the single mode approximation (SMA) [ 90]. Therehas recently been rapid development in the study of quantum entanglement delity in non-inertial frames that goesbeyond this approximation. For instance, it has been shown that beyond SMA and for some choices of the bipartitestate, the accessible entanglement for an accelerated observer may behave in a non-monotonic way, conversely to therst results reported in Refs. [ 87, 88]. This is due to inaccessible correlations in the initial states becoming accessible tothe accelerated observer when his proper Fock basis changes as acceleration varies [ 91].

Furthermore, while the idealized quantum eld modes used in previous studies are arguably impossible to reproducein any experimental setting (due to their non-localized and highly oscillating spatial proles [ 90]), the study of thebehaviour of localized eld states has just recently started. Both localized projective measurements [ 92] and homodynedetection schemes [93] have the edge on other idealized scenarios for experiments where these effects can be detected.Other localized experimental settings where entanglement

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